Generalized Volterra-type integral operators between Bloch-type spaces

The Volterra-type integral operator plays an essential role in modern complex analysis and operator theory. Recently, Chalmoukis \cite{Cn} introduced a generalized integral operator, say $I_{g,a}$, defined by $$I_{g,a}f=I^n(a_0{f}^{(n-1)}{g}^{\mathrm{\prime }}+a_1{f}^{(n-2)}{g}^{\mathrm{\prime }\mathrm{\prime }}+\cdots +a_{n-1}f{g}^{(n)}),$$ where $g\in H(\mathbb{D})$ and $a=(a_0,a_1,\cdots ,a_{n-1})\in {\mathbb{C}}^n$. $I^n$ is the $n$th iteration of the integral operator $I$. In this paper, we introduce a more generalized integral operators $I_{\mathbf{g}}^{(n)}$ that cover $I_{g,a}$ on the Bloch-type space ${\mathcal{B}}^{\alpha }$, defined by $$I_{\mathbf{g}}^{(n)}f=I^n(f{g}_0+\cdots +f^{(n-1)}{g}_{n-1})\mathrm{.}$$ We show the rigidity of the operator $I_{\mathbf{g}}^{(n)}$ and further the sum ${\sum }_{i=1}^n{I}_{g_i}^{N_i,k_i}$, where $I_{g_i}^{N_i,k_i}f=I^{N_i}(f^{(k_i)}{g}_i)$. Specifically, the boundedness and compactness of ${\sum }_{i=1}^n{I}_{g_i}^{N_i,k_i}$ are equal to those of each $I_{g_i}^{N_i,k_i}$. Moreover, the boundedness and compactness of $I^n((f{g}^{\mathrm{\prime }}{)}^{(n-1)})$ are independent of $n$ when $\alpha >1$.